Multiresolutional Analysis 327 the solution x and d,is its fine-scale component.Formally eliminating d,from (7.5)by substituting d.=Asd-Bs s.)yields Ts,-Cs As Bs,)s,=s1-Cs,As dr (7.7) This equation is called a reduced equation,while the operator Rs,=Ts,-Cs As,Bs (7.8) is a one step reduction of the operator S;also known as the Schur complement of As the block matrix Bsy Cs,Ts, Note that the solution s,of the reduced equation is exactly P,where P is the projection onto V andx is the solution of (7.4).Note that the reduced equation is not the same as the averaged equation,which is given by Ts 5:=51 (7.9) Once we have obtained the reduced equation,it may formally be reduced again to produce an equation on Vand the solution of this equation is just the V2 component of the solution for (7.4).Likewise,we may reduce these equations recursively n times (assuming that,if the multiresolution analysis is on a bounded domain,then j+ns)to produce an equation on V the solution of which is the projection of the solution of (7.4)on V We note that in the finite-dimensional case,if we are considering a multiresolution analysis defined on a domain in R,the reduced equation(7.5)has half as many unknowns as the original equation(7.4).If the domain is in R2,then the reduced equations have one-fourth as many unknowns as the original equation. Reduction,therefore,preserves the coarse-scale behaviour of solutions while reducing the number of unknowns. In order to iterate the reduction step over many scales,we need to preserve the form of the equation as a way of deriving a recurrence relation.In (7.4)and(7.5), both S,and Rs,are matrices,and thus the procedure may be repeated.However, identification of the matrix structure is usually not sufficient.In particular,even though the matrix A for ODEs and PDEs is sparse,the component As term may become dense,changing the equation from a local one to a global one.It is important to know under what circumstances the local nature of the differential
Multiresolutional Analysis 327 the solution x and x d is its fine-scale component. Formally eliminating x d from (7.5) by substituting ( ) x S f S x d A d B s j j = − −1 yields ( ) S S S S x f CS AS d f T C A B s s j j j j j j −1 −1 − = − (7.7) This equation is called a reduced equation, while the operator S j S j S j S j S j R T C A B−1 = − (7.8) is a one step reduction of the operator S j also known as the Schur complement of the block matrix ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ j j j j S S S S C T A B . Note that the solution x s of the reduced equation is exactly P x j+1 , where Pj+1 is the projection onto Vj+1 and x is the solution of (7.4). Note that the reduced equation is not the same as the averaged equation, which is given by S x f T s s j = ~ (7.9) Once we have obtained the reduced equation, it may formally be reduced again to produce an equation on Vj+2 and the solution of this equation is just the Vj+2 component of the solution for (7.4). Likewise, we may reduce these equations recursively n times (assuming that, if the multiresolution analysis is on a bounded domain, then j+n§0) to produce an equation on Vj+n , the solution of which is the projection of the solution of (7.4) on Vj+n . We note that in the finite-dimensional case, if we are considering a multiresolution analysis defined on a domain in R, the reduced equation (7.5) has half as many unknowns as the original equation (7.4). If the domain is in R2 , then the reduced equations have one-fourth as many unknowns as the original equation. Reduction, therefore, preserves the coarse-scale behaviour of solutions while reducing the number of unknowns. In order to iterate the reduction step over many scales, we need to preserve the form of the equation as a way of deriving a recurrence relation. In (7.4) and (7.5), both S j and S j R are matrices, and thus the procedure may be repeated. However, identification of the matrix structure is usually not sufficient. In particular, even though the matrix A for ODEs and PDEs is sparse, the component −1 j AS term may become dense, changing the equation from a local one to a global one. It is important to know under what circumstances the local nature of the differential
328 Computational Mechanics of Composite Materials operator may be(approximately)preserved.Furthermore,if the equation is of the form of -V(e(x)Vu(x))=f(x) (7.10) or some other variable-coefficient differential equation,we should verify if the reduction procedure preserves this form,so that we may find effective coefficients of the equation on the coarse scale.This process is the basic goal of homogenisation techniques,and it extracts information from the reduced equation based on the form of the original equation.Thus,within the multiresolution approach,reduction and homogenisation are closely related but have different goals:homogenisation attempts to find effective equations and their coefficients on the coarse scale,whereas reduction merely finds a coarse-scale version of a given system of equations. The multiresolutional (MRA)homogenisation procedure is applied to the systems of ODEs,which may be written in the form Bx+q+入=K(Ax+p) (7.11) In particular,we consider equations of the form (+Bx(t)+gt)+元=∫A(s)x(s)+p(s)ds,t∈(0,1) (7.12) 0 on L(0,1),where B(t)and A(t)are n x n matrix-valued functions,p(t)and g(t)are vector forcing terms,and x(t)is the solution vector.As a differential equation this is written as +B00+g0）=A0a0+p0 (7.13) with the initial conditions (I+B(0)x(0)=-g(0)-.On V,,j<0,the projection of (7.11)is written as B+9+元=K,4x,+Pj (7.14) or Sx;=fi (7.15) where S,=B-K,A,f=KjP1-4-九，x=Px (7.16
328 Computational Mechanics of Composite Materials operator may be (approximately) preserved. Furthermore, if the equation is of the form of −∇( ) e(x)∇u(x) = f (x) (7.10) or some other variable-coefficient differential equation, we should verify if the reduction procedure preserves this form, so that we may find effective coefficients of the equation on the coarse scale. This process is the basic goal of homogenisation techniques, and it extracts information from the reduced equation based on the form of the original equation. Thus, within the multiresolution approach, reduction and homogenisation are closely related but have different goals: homogenisation attempts to find effective equations and their coefficients on the coarse scale, whereas reduction merely finds a coarse-scale version of a given system of equations. The multiresolutional (MRA) homogenisation procedure is applied to the systems of ODEs, which may be written in the form Bx + q + λ = K( ) Ax + p (7.11) In particular, we consider equations of the form () ( ) + + + = ∫ + t I B t x t q t A s x s p s ds 0 ( ) ( ) ( ) λ ( ) ( ) ( ) , ) t ∈ (0,1 (7.12) on ) (0,1 2 L , where B(t) and A(t) are n x n matrix-valued functions, p(t) and q(t) are vector forcing terms, and x(t) is the solution vector. As a differential equation this is written as ( ) ( ) I B(t) x(t) q(t) A(t)x(t) p(t) dt d + + = + (7.13) with the initial conditions ( ) I + B(0) x(0) = −q(0) − λ . On Vj , j<0, the projection of (7.11) is written as ( ) j j j j j j p j B x + q + λ = K A x + (7.14) or j j j S x = f (7.15) where S j = Bj − K j Aj , f j = K j p j − q j − λ , j j j x = P x (7.16)
Multiresolutional Analysis 329 After a single reduction,our goal is to have an equation on V of the form B阳x州+9州+元=K49州+P阳) (7.17) where=P B=P etc.We use the notation B(to indicate that the equation is first projected to a scale V,,and then the reduction procedure is applied /-j times to obtain an equation on V.This notation therefore indicates that (7.17)was obtained by a single reduction of the same form of equation on V;one time to produce an equation on the coarser scale V. It allows one to establish a recurrence relation for k=jj+1,...,0 between the operators and forcing terms Bpon V.It turns out that this task of finding the recurrence relations is simplified significantly if one uses a multiresolution analysis whose basis functions have non-overlapping support.We use the Haar basis,but a multiwavelet basis may be used if higher order elements are necessary. In the Haar basis,the operators B,A and K derived from equations of the form of(7.14)have a simple form.Each of these is in an (Nn)x(Nn)matrix, where N=2 is the number of unknowns on the scale V,and n denotes the number of equations in the original system.Furthermore,B and A are both block- diagonal matrices.The diagonal blocks of B and A are n x n matrices.There are therefore N,diagonal blocks,each of which is an n x n matrix.For B and A we denote their ith diagonal blocks by (e,】na(a The matrices are given by the Haar coefficients of the n x n matrix-valued functions B(x)and A(x)on the scale V,.It can be written that B,=dag{1+B,'-1 (7.18) and 4,=daga,)- (7.19) where
Multiresolutional Analysis 329 After a single reduction, our goal is to have an equation on Vj+1 of the form ( ) ( ) 1 ( ) 1 ( ) 1 1 ( ) 1 ( ) 1 ( ) 1 j j j j j j j j j j j j j B + x + + q + + λ = K + A + x + + p + (7.17) where j j j j x P x1 ( ) +1 = + , j j j j B P x1 ( ) +1 = + , etc. We use the notation ( j) Bl to indicate that the equation is first projected to a scale Vj , and then the reduction procedure is applied l-j times to obtain an equation on Vl . This notation therefore indicates that (7.17) was obtained by a single reduction of the same form of equation on Vj one time to produce an equation on the coarser scale Vj+1 . It allows one to establish a recurrence relation for k=j,j+1,...,0 between the operators and forcing terms ( ) ( ) ( ) ( ) , , , j k j k j k j k B x p q on Vk+1 . It turns out that this task of finding the recurrence relations is simplified significantly if one uses a multiresolution analysis whose basis functions have non-overlapping support. We use the Haar basis, but a multiwavelet basis may be used if higher order elements are necessary. In the Haar basis, the operators j j B , A and j K derived from equations of the form of (7.14) have a simple form. Each of these is in an (Njn) x (Njn) matrix, where Nj=2j is the number of unknowns on the scale Vj and n denotes the number of equations in the original system. Furthermore, j B and j A are both blockdiagonal matrices. The diagonal blocks of j B and j A are n x n matrices. There are therefore Nj diagonal blocks, each of which is an n x n matrix. For j B and j A we denote their ith diagonal blocks by i j B ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ and i j A ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ . The matrices are given by the Haar coefficients of the n x n matrix-valued functions B(x) and A(x) on the scale Vj . It can be written that { } ( ) 2 1 0 = − = = + j i i i j Bj B diag I (7.18) and { } ( ) 2 1 0 = − = = j i i i j Aj A diag (7.19) where
330 Computational Mechanics of Composite Materials 0 0 0 1 0 0 K,=δ 1 (7.20) 4 1 where is the identity matrix.andB and are the h Haar coefficients on scale V;of the n x n matrix-value functions B(x)as well as A(x).For(7.17),the recursion relations are given by =6,-D,r-o,+9, (7.21) ()--0--6))6 (7.22) =6)-DFD,)+6,》 (7.23) %,=6,-o,)-(D,-6F-o,-s,》 (7.24) where Sa,=49+a9),(D,=4b-(49h) (7.25) SB,=(B”+(BPh)(Ds),=BPA-(Bb） (7.26) sn)=古ph+(ph,(D,=方ph-(p） (7.27) (S,】=方aPa+gP）.(D,)=方aPh-gPh） (7.28) and,finally, F=1+5+受D (7.29) Note that the recurrence relations are local and can be carried out over many scales as needed (assuming the existence of F at each scale).Starting with (7.17)on V and,reducing jtimes.yields on V Bx8”+q8+=KAx+p） (7.30)
330 Computational Mechanics of Composite Materials ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = I I I I I I I I I K j j 2 1 2 1 2 1 2 1 ... ... ... ... 0 ... 0 0 0 ... 0 δ (7.20) where j j − δ = 2 , I is the n x n identity matrix, and i j B ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ and i j A ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ are the ith Haar coefficients on scale Vj of the n x n matrix-value functions B(x) as well as A(x). For (7.17), the recursion relations are given by ( ) ()( ) ( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − + − + A i k i A i A i B i j Ak S D F D S 2 ( ) 1 1 δ (7.21) ( ) ( ) () () () ( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ + ⎠ ⎞ ⎜ ⎝ ⎛ = − − − − + A i k A i B i k A i B i k i B i j Bk S D D S F D S 2 2 2 ( ) 1 1 δ δ δ (7.22) ( ) ( ) ( ) ( ) () ()i p i A i q k i i p j k p = S − D F D + S − + ( ) 1 1 2 δ (7.23) ( ) ( ) ( ) ( ) ( )() ( ) ()()i p i B i A i q k i p k i i q j k q = S − D − D − S F D − S − + ( ) 1 1 2 2 δ δ (7.24) where ( ) ( ) ()()2 1 ( ) 2 ( ) 2 1 + = + i j i k j S A i Ak A , ( ) ( ) ()()2 1 ( ) 2 ( ) 2 1 + = − i j i k j DA i Ak A (7.25) ( ) ( ) ()()2 1 ( ) 2 ( ) 2 1 + = + i j i k j S B i Bk B , ( ) ( ) ()()2 1 ( ) 2 ( ) 2 1 + = − i j i k j DB i Bk B (7.26) ( ) ( ) ()()2 1 ( ) 2 ( ) 2 1 + = + i j i k j k i S p p p , ( ) ( ) ()()2 1 ( ) 2 ( ) 2 1 + = − i j i k j k i Dp p p (7.27) ( ) ( ) ()()2 1 ( ) 2 ( ) 2 1 + = + i j i k j k i Sq q q , ( ) ( ) ()()2 1 ( ) 2 ( ) 2 1 + = − i j i k j k i Dq q q (7.28) and, finally, ( ) ( ) A i k F I S B i D 2 δ = + + (7.29) Note that the recurrence relations are local and can be carried out over many scales as needed (assuming the existence of −1 F at each scale). Starting with (7.17) on V− j and, reducing j times, yields on V0 ( ) ( ) 0 ( ) 0 ( ) 0 0 ( ) 0 ( ) 0 ( ) 0 j j j j j j B x + q + λ = K A x + p (7.30)
Multiresolutional Analysis 331 where to compute B,Ap we use the recurrence relations jtimes. Multiresolutional homogenisation is formulated as follows.First,we consider the limit of (7.30)as j,therefore B6x6}+q6+元=K46x6+p） (7.31) It is employed to eliminate infinite number of fine scales from the original equation.The matrices BA are called the reduced coefficients of (7.14). Then,we look for the operators and forcing terms B"(t),A"(t),p"(t),g"(t)with certain desired qualities(e.g.constant values)such that the equation k()+q")+=f(A"(s)x(s)+p"(s))ds.tE(O.D) (7.32) subjected to the same reduction and limit procedure as (7.12),yields on V the same equation as in (7.31).For (7.12)we usually require that B",A",p""be constant.The result of homogenisation in this case is summarised as follows: Theorem Given(7.12),if the limits,which determine the matrices B and A exist, then there exist constant matrices B,A and forcing terms p",",such that the reduced coefficients and forcing terms of (7.32)are given by BAp.The homogenised coefficients B".A"and forcing terms p",q"are defined by A=A (7.33) B=AA-1-1 (7.34) p*=p (7.35) =6--i-广p (7.36) where A=1og+(0+B6)-AA (7.37) Proof It is observed that for the constant coefficients the recurrence relations (7.21)and (7.22)simplify to
Multiresolutional Analysis 331 where to compute ( ) 0 ( ) 0 ( ) 0 ( ) 0 , , , j j j j B A p q we use the recurrence relations j times. Multiresolutional homogenisation is formulated as follows. First, we consider the limit of (7.30) as j → −∞ , therefore ( ) ( ) 0 ( ) 0 ( ) 0 0 ( ) 0 ( ) 0 ( ) 0 −∞ −∞ −∞ −∞ −∞ −∞ B x + q + λ = K A x + p (7.31) It is employed to eliminate infinite number of fine scales from the original equation. The matrices ( ) 0 ( ) 0 , −∞ −∞ B A are called the reduced coefficients of (7.14). Then, we look for the operators and forcing terms ) B (t), A (t), p (t), q (t h h h h with certain desired qualities (e.g. constant values) such that the equation () ( ) + + + = ∫ + t h h h h I B t x t q t A s x s p s ds 0 ( ) ( ) ( ) λ ( ) ( ) ( ) , ) t ∈ (0,1 (7.32) subjected to the same reduction and limit procedure as (7.12), yields on V0 the same equation as in (7.31). For (7.12) we usually require that h h h h B , be A , p , q constant. The result of homogenisation in this case is summarised as follows: Theorem Given (7.12), if the limits, which determine the matrices ( ) 0 −∞ B and ( ) 0 −∞ A exist, then there exist constant matrices h h B , and forcing terms A h h p , , such that the q reduced coefficients and forcing terms of (7.32) are given by ( ) 0 ( ) 0 ( ) 0 ( ) 0 , , , −∞ −∞ −∞ −∞ B A q p . The homogenised coefficients h h B , and forcing terms A h h p , are defined by q ( ) 0 −∞ A = A h (7.33) B A A I h h = − ~ −1 (7.34) ( ) 0 −∞ p = p h (7.35) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + − − − − − h −∞ h h q q I A A A I A p 1 1 2 ( ) 1 0 ~ exp ~ ~ (7.36) where ( ( ) ) h h A I I B A A 1 2 ( ) 1 0 log ~ − −∞ = + + − (7.37) Proof It is observed that for the constant coefficients the recurrence relations (7.21) and (7.22) simplify to